# How is The strengthened finite Ramsey theorem known to be true for natural numbers?

http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem states that independence of "The strengthened finite Ramsey theorem" from PA is proved by implying consistency of PA. But how do we know that it's actually true for the Model that's Natural Numbers. How was it even proved for Natural Numbers? Under what logical system is it proved, if not for Peano's Axioms? And how do we know that such a logical system on which it's proved actually models Natural Numbers?

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No, that article doesn't state that. That doesn't even make sense. –  Chris Eagle Oct 21 '12 at 11:44
ha, i meant the "independence" of the theorem is proved by implying the consistency of PA. Editing the question now. Thanks! –  rajeshsr Oct 21 '12 at 13:46